The zero forcing number Z ( G ) of a graph G was proposed by the AIM Minimum Rank-Special Graphs Work Group as an upper bound on the nullities of matrices associated with G . Recently, the study of upper bounds for the zero forcing number and for the nullity of a connected graph in terms of its order and maximum degree has received much attention. In particular, Gentner and Rautenbach (2018) proved that if G is a connected graph of order n with maximum degree Δ ≥ 3 , then Z ( G ) ≤ Δ − 2 Δ − 1 n except when G is a complete graph, a complete bipartite graph of the form K n 1 , n 2 with | n 1 − n 2 | ≤ 1 , or is equal to W 1 , W 2 , where W 1 and W 2 are two specific graphs of order 5 and 7, respectively. In this paper we identify all connected graphs G of order n with maximum degree Δ ≥ 3 that satisfy Z ( G ) = Δ − 2 Δ − 1 n , and prove that if Z ( G ) < ( Δ − 2 ) n Δ − 1 then Z ( G ) ≤ ( Δ − 2 ) n − 1 Δ − 1 . We find one graph missing from the list of exceptional graphs in the above-mentioned result of Gentner and Rautenbach and provide an independent alternative proof for the amended result. A new proof technique which is based on the concept of maximal augmenting path is introduced in the course of proofs. We also rederive or improve existing upper bounds for the nullity of a connected graph in terms of its order and maximum degree.

This article investigates the problem of synchronous randomized neighbor discovery with directional antennas. Due to the long tail effect, it will take long time to discover the last few neighbors, which increases overall neighbor discovery time. This effect is due to small proportion of remaining undiscovered neighbors. Moreover, improper choices of reception probabilities make the discovery even worse. In this article, a cognitive framework is proposed to minimize the expectation of neighbor discovery time. We present a scheme in which reception probabilities are dynamically adjusted. We consider an ideal scenario and a practical scenario. In an ideal scenario where perfect information about the number of neighbors is available, reception probabilities are adjusted according to the number of neighbors. A method of dynamic programming is used to recursively calculate the optimal reception probabilities. In an actual scenario where perfect information about number of neighbors is unavailable, a neighbor estimation method based on maximum-likelihood estimation is executed before probability adjustment. Simulation results show that when perfect information about neighbor is available and total transmission probability is within a proper range (between 0.1 and 0.2), the average neighbor discovery time can be significantly reduced (by 38% to 43%, respectively) compared with an existing probability-fixed scheme. With imperfect information, the scheme also works well and realizes appreciable reduction in average neighbor discovery time compared with existing self-adaptive schemes.

In this study, the air–ground integrated deployment method is studied for the unmanned aerial vehicle (UAV)‐enabled mobile edge computing (MEC) system. The UAV can help to reduce the delay and energy consumption of MEC. However, the limited coverage range of UAV limits the quality of data offloading. To improve the efficiency of data transmission, a hierarchical game model is designed. Ground nodes form multiple coalitions actively according to the position of UAV and the UAV adjusts the position based on the data distribution of ground networks. The relationship between the UAV and ground nodes is modelled as a Stackelberg game. A coalition formation game (CFG) is constructed for the data gathering among ground nodes. It is proved that the proposed CFG is an exact potential game with at least one Nash equilibrium. Moreover, the property of Stackelberg equilibrium is proven for the air–ground cooperative relationship. Based on the hierarchical model, a distributed air–ground integrated deployment algorithm is proposed to jointly optimise the position of the UAV and the coalition formation of ground nodes. The simulation results show that the proposed method promotes the efficiency of data transmission greatly and can converge to a stable state with reasonable iteration times.